Mathbox for Jarvin Udandy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  plvcofph Structured version   Visualization version   GIF version

Theorem plvcofph 43707
 Description: Given, a,b,d, and "definitions" for c, e, f: f is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
Hypotheses
Ref Expression
plvcofph.1 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
plvcofph.2 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
plvcofph.3 (𝜂 ↔ (𝜒𝜏))
plvcofph.4 𝜑
plvcofph.5 𝜓
plvcofph.6 𝜃
Assertion
Ref Expression
plvcofph 𝜂

Proof of Theorem plvcofph
StepHypRef Expression
1 plvcofph.1 . . . 4 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
2 plvcofph.4 . . . 4 𝜑
3 plvcofph.5 . . . 4 𝜓
41, 2, 3plcofph 43705 . . 3 𝜒
5 plvcofph.2 . . . 4 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
6 plvcofph.6 . . . 4 𝜃
75, 2, 3, 4, 6pldofph 43706 . . 3 𝜏
84, 7pm3.2i 474 . 2 (𝜒𝜏)
9 plvcofph.3 . . . 4 (𝜂 ↔ (𝜒𝜏))
109bicomi 227 . . 3 ((𝜒𝜏) ↔ 𝜂)
1110biimpi 219 . 2 ((𝜒𝜏) → 𝜂)
128, 11ax-mp 5 1 𝜂
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator